This suppression of PDI by the flux-tube expansion factor (\(f_\mathrm {max}\) ) can be explained by considering the plasma beta (Figure REF ). Here, the plasma beta (\(\beta \) ) is time-averaged, as defined in Equation REF  and plotted for pump frequencies \(0.3,~1, and~3~\) mHz. The plasma beta is defined as \(\beta = P/(B^2/2\mu _0)\) , where \(P\)  is the total thermal pressure and \(B\)  is the magnitude of the total magnetic field. In the absence of PDI, we observe the \(\beta \ll 1\)  conditions as the solar wind expands (Figure  REF (a)). The subsequent large-amplitude fluctuations generated by the PDI instability increase the plasma beta as seen for \(f_\mathrm {max} = 3\)  in Figure REF (b) and \(f_\mathrm {max} = 3,~5,~10\)  in Figure REF (c). Upon comparing Figures REF (a) & REF (c), we see that plasma beta increases from \(<0.1\)  to \(~2\)  for \(f_\mathrm {max} = 10\) , but only from \(\ll 0.1\)  to \(\approx 0.2\)  for \(f_\mathrm {max} = 3\) . Thus, the plasma beta is small when \(f_\mathrm {max}\)  is lower, and the relative increase in \(\beta \)  is smaller for a lower \(f_\mathrm {max}\)  when the PDI is excited. Previous analytical studies [1]}, [2]} have shown the inverse dependence of the PDI growth rate on plasma beta (\(\gamma _\mathrm {PDI} \propto \beta ^{-1/4}\) ) in the limit \(\beta \ll 1\) . Considering the decrease in \(\beta \)  as the solar wind expands more slowly (smaller \(f_\mathrm {max}\) ), the onset of PDI is suppressed as the magnetic flux tube expansion factor increases. This variation in the PDI growth rate with \(f_\mathrm {max}\)  is shown by [3]}.
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